A counterexample to Kaplansky’s zero-divisor conjecture

1 · The essentials

Multiplying two nonzero objects can sometimes give zero, even in a carefully structured mathematical system. The manuscript constructs such an example in a system built from a group, which is a collection of elements with a rule for combining them. In this group, repeatedly combining any element other than the identity with itself never reaches the identity. Nevertheless, when finite sums of group elements are allowed, with coefficients calculated modulo two, two nonzero sums can multiply to zero. This contradicts the prediction known as Kaplansky’s zero-divisor conjecture. The main counterexample has a Lean formalization. The manuscript also states that the group has a finite description and a finite two-dimensional geometric model of a particular kind.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine two nonempty sets of instructions whose combined effects cancel completely. The example has a similar contrast: neither algebraic input is zero, but their product is zero; this image does not capture the precise rules of multiplication.

Why it matters

The counterexample shows that the stated restriction on the group is not enough to prevent nonzero zero divisors in its group algebra. It also comes with additional finiteness properties: the group is finitely presented and admits a finite two-dimensional classifying space.

2 · Going further

The problem concerns a group algebra $F_{2} [G]$ : its elements are finite formal sums of elements of $G$ , with coefficients in the two-element field $F_{2}$ . Multiplication extends the group operation, and coefficients are calculated modulo two. Nonzero zero divisors are elements $a, b \neq = 0$ such that $ab = 0$ . The conjecture at issue predicts the absence of such pairs when the underlying group is torsion-free.

The main manuscript constructs a finitely presented torsion-free group $G$ for which such a pair exists in $F_{2} [G]$ . Torsion-free means that no nonidentity element has finite order. Finite presentation means that the group can be specified by finitely many generators and finitely many defining relations. The manuscript additionally states that $G$ admits a finite two-dimensional classifying space.

The abstract describes the approach as an explicit construction of a counterexample, but supplies neither the group presentation nor the zero divisors or the details establishing torsion-freeness. The main result has a Lean formalization; the supplied description does not specify whether that formalization covers every additional claim. Only one manuscript is supplied, so no relationship to companion manuscripts can be described.

Useful notions

Groups and the identity element
Finite order and torsion-free groups
Arithmetic modulo two
Group algebras and zero divisors

What was known before / what is new

Before

The source identifies Kaplansky’s zero-divisor conjecture as the conjecture being contradicted. It supplies no account of earlier partial results or the historical development of the problem.

New

The manuscript constructs a finitely presented torsion-free group whose group algebra over $F_{2}$ contains nonzero zero divisors. It also states that the group has a finite two-dimensional classifying space. The main counterexample has a Lean formalization.

Limits and assumptions

The stated example uses coefficients in $F_{2}$ and a particular constructed torsion-free group; the source gives no corresponding claims for other coefficient fields or other groups. The abstract does not provide the construction or its supporting argument. The main result is Lean-formalized, but the supplied material does not delineate the formalization’s coverage of the additional finiteness and geometric claims. No remaining open problems are specified.

Mini glossary

Torsion-free group: A group in which no element other than the identity has finite order.
Group algebra: An algebra of finite formal sums of group elements with coefficients in a field, whose multiplication extends the group operation.
Nonzero zero divisors: Two nonzero algebra elements whose product is zero.
Finitely presented group: A group described by finitely many generators and finitely many defining relations.
$F_{2}$: The field with two elements, using arithmetic modulo two.

Keywords : Group algebras · Zero divisors · Torsion-free groups

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Constructs a finitely presented torsion-free group G whose group algebra $F_{2} [G]$ has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/196.md

Manuscripts in this family

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A Torsion-Free Group Algebra with Zero Divisors

23/09/2026

Verified by Lean

We disprove Kaplansky's zero-divisor conjecture by constructing a finitely presented torsion-free group G for which $F_{2} [G]$ has nonzero zero divisors. The group admits a finite two-dimensional classifying space.

PDF GitHub folder Lean formalization